For the case of \(d\le 2\) , where every \(\pi _\sigma (\cdot )\)  is recurrent, we will define the multi-layer SEP process on the ladder graph (see e.g. [1]}), i.e., we define the state space \(\Omega _{-1}^{\prime } := \lbrace 0,1\rbrace ^{V\times A}\)  with \(A = \lbrace 1, 2,..., \alpha \rbrace \) . This space can be seen as the space where on every site \(v \in V\)  there is a ladder with \(\alpha \)  steps, and every particle chooses a step of this ladder if it moves to a new site. We can easily go back from a configuration \(\eta ^{\prime } \in \Omega _{-1}^{\prime }\)  to a configuration in \(\eta \in \Omega _{-1}\)  by setting
\(\eta (x,\sigma ) = \sum _{i=1}^\alpha \eta ^{\prime }(x,\sigma ,i),\ \ \ \ \text{for all $(x,\sigma ) \in V$}.\) 
